Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on Scalar curvature comparison rigidity for compact domains

Xuan Yao

Abstract

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, which is a parallel of Shi-Tam's results.

A Note on Scalar curvature comparison rigidity for compact domains

Abstract

We prove a generalization of Gromov's conjecture on scalar curvature rigidity of convex polytopes to arbitrary convex Riemannian polytope type domains via harmonic spinors on convex domians with boundary condition constructed by Brendle. In particular, we prove a rigidity results on comparison of scalar curvature and scaled mean curvature on the boundary for any convex domain in Euclidean space, which is a parallel of Shi-Tam's results.

Paper Structure

This paper contains 8 sections, 15 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Acknowledgements
  4. Prelimaries
  5. Definitions and Notations
  6. Spinor method
  7. Morrey norm estimates
  8. Further discussions

Key Result

Theorem 1.3

Suppose $\Omega^n$ is a convex Riemannian polytope type domain in $\mathbb{R}^n$, $n\geq 3$. Assume that $g$ is a Riemannian metric on $\Omega$ and it satisfies: Then the metric $g$ must be Euclidean.

Theorems & Definitions (39)

  • Conjecture 1.1: Gromov gromov2014dirac
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Matching Angle Hypothesis
  • Proposition 2.5: Brendle brendle2023matchingangle
  • Proposition 2.6: Brendle brendle2023matchingangle
  • ...and 29 more